Chord-arc Curves Research Articles

Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations

AbstractWe present an α‐regularization of the Birkhoff‐Rott equation (BR‐α equation), induced by the two‐dimensional Euler‐α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler‐α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR‐α equation, remains Lipschitz for all times, (ii) an initially Hölder C1, β, 0 ≤ β < 1, chord arc curve remains in C1, β for all times, and finally, (iii) an initially Hölder Cn, β, n ≥ 1, 0 < β < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler‐α and the BR‐α descriptions of the vortex sheet motion are equivalent. © 2009 Wiley Periodicals, Inc.

Higher Dimensional Ahlfors Regular Sets and Chordarc Curves inR n

Higher Dimensional Ahlfors Regular Sets and Chordarc Curves inR n

A regular curve that cannot be covered with finitely many chord-arc curves

A regular curve that cannot be covered with finitely many chord-arc curves

Quasiconformal mappings and chord-arc curves

Given a quasiconformal mapping ρ \rho on the plane, what conditions on its dilatation μ \mu guarantee that ρ ( R ) \rho ({\mathbf {R}}) is rectifiable and ρ | R \rho {|_{\mathbf {R}}} is locally absolutely continuous? We show in this paper that if μ \mu satisfies certain quadratic Carleson measure conditions, with small norm, then ρ ( R ) \rho ({\mathbf {R}}) is a chord-arc curve with small constant, and ρ ( x ) = ρ ( 0 ) + ∫ 0 x e a ( t ) d t \rho (x) = \rho (0) + \int _0^x {{e^{a(t)}}dt} for x ∈ R x \in {\mathbf {R}} , with a ∈ BMO a \in \operatorname {BMO} having small norm. Conversely, given any such map from R → C {\mathbf {R}} \to {\mathbf {C}} , we show that it has an extension to C {\mathbf {C}} with the right kind of dilatation. Similar results hold with R {\mathbf {R}} replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.

Estimates for (\overline∂-𝜇∂)⁻¹ and Calderón’s theorem on the Cauchy integral

One can view the Cauchy integral operator as giving the solution to a certain ∂ ¯ \overline \partial problem. If one has a quasiconformal mapping on the plane that takes the real line to the curve, then this ∂ ¯ \bar \partial problem on the curve can be pulled back to a ∂ ¯ − μ ∂ \bar \partial - \mu \partial problem on the line. In the case of Lipschitz graphs (or chordarc curves) with small constant, we show how a judicial choice of q.c. mapping and suitable estimates for ∂ ¯ − μ ∂ \bar \partial - \mu \partial gives a new approach to the boundedness of the Cauchy integral. This approach has the advantage that it is better suited to related problems concerning H ∞ {H^\infty } than the usual singular integral methods. Also, these estimates for the Beltrami equation have application to quasiconformal and conformal mappings, taken up in a companion paper.

A counterexample in conformal welding concerning chord-arc curves

A counterexample in conformal welding concerning chord-arc curves

Cauchy Integral of Calderón on the Graphs of Functions with BMO Derivatives

AbstractWe first note that each graph (x,A(x)) of a function A(x) with BMO derivative is a chord-arc curve. Using this, Muckenhoupt's Ap theory, and the theory of Calderón-Zygmund operators, we shall derive weighted norm inequalities for the Cauchy integral on such graphs from a recent theorem of G. David on the L2-boundedness of Cauchy integral on almost-lipschitzian curves.